Schild's ladder


In the theory of general relativity, and differential geometry more generally, Schild's ladder is a first-order method for approximating parallel transport of a vector along a curve using only affinely parametrized geodesics. The method is named for Alfred Schild, who introduced the method during lectures at Princeton University.

Construction

The idea is to identify a tangent vector x at a point with a geodesic segment of unit length, and to construct an approximate parallelogram with approximately parallel sides and as an approximation of the Levi-Civita parallelogramoid; the new segment thus corresponds to an approximately parallel translated tangent vector at
Formally, consider a curve γ through a point A0 in a Riemannian manifold M, and let x be a tangent vector at A0. Then x can be identified with a geodesic segment A0X0 via the exponential map. This geodesic σ satisfies
The steps of the Schild's ladder construction are:
This is a discrete approximation of the continuous process of parallel transport. If the ambient space is flat, this is exactly parallel transport, and the steps define parallelograms, which agree with the Levi-Civita parallelogramoid.
In a curved space, the error is given by holonomy around the triangle which is equal to the integral of the curvature over the interior of the triangle, by the Ambrose-Singer theorem; this is a form of Green's theorem, and in the case of Levi-Civita connections on surfaces, of Gauss–Bonnet theorem.